Shape-sensitive crystallization in colloidalsuperball fluidsLaura Rossia,b,1,2,3, Vishal Sonia,2,3, Douglas J. Ashtonc,2, David J. Pined, Albert P. Philipseb, Paul M. Chaikind,Marjolein Dijkstrac, Stefano Sacannae, and William T. M. Irvinea,3

aJames Franck institute, Department of Physics, University of Chicago, Chicago, IL 60637; bVan’t Hoff Laboratory for Physical and Colloid Chemistry, DebyeInstitute for Nano-materials Science, Utrecht University, 3584 CH Utrecht, The Netherlands; cSoft Condensed Matter, Debye Institute for Nano-materialsScience, Utrecht University, 3584 CC Utrecht, The Netherlands; and dCenter for Soft Matter Research, Department of Physics, and eMolecular DesignInstitute, Department of Chemistry, New York University, New York, NY 10003

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved March 16, 2015 (received for review August 11, 2014)

Guiding the self-assembly of materials by controlling the shape ofthe individual particle constituents is a powerful approach tomaterial design. We show that colloidal silica superballs crystallizeinto canted phases in the presence of depletants. Some of thesephases are consistent with the so-called “Λ1” lattice that was re-cently predicted as the densest packing of superdisks. As the sizeof the depletant is reduced, however, we observe a transition to asquare phase. The differences in these entropically stabilizedphases result from an interplay between the size of the depletantsand the fine structure of the superball shape. We find qualitativeagreement of our experimental results both with a phase diagramcomputed on the basis of the volume accessible to the depletantsand with simulations. By using a mixture of depletants, one ofwhich is thermosensitive, we induce solid-to-solid phase transi-tions between square and canted structures. The use of depletantsize to leverage fine features of the shape of particles in drivingtheir self-assembly demonstrates a general and powerful mecha-nism for engineering novel materials.

superballs | phase behavior | dense packings | depletion interactions |Monte Carlo simulations

Determining the relationship between the macroscopic struc-ture of a material and the properties of its microscopic con-

stituents is a fundamental problem in condensed matter science.A particularly interesting aspect of this problem is to understandhow the self-assembly of a collection of particles is determinedby their shape. These so-called “packing problems” have longinterested physicists, mathematicians, and chemists alike andhave been used to understand the structures of many condensedphases of matter (1–3). Computational and experimental ad-vances continue to enable new explorations into fundamentalaspects of these problems today (4–13). Recent discoveries in-clude dense packings of tetrahedra into disordered, crystalline,and quasi-crystalline structures (14, 15), as well as the singulardense packings of ellipsoids (16).Technologically speaking, these discoveries are becoming in-

creasingly crucial as new synthesis techniques are allowing forthe creation of more and more complex shaped nanoscopic andmicroscopic particles (17, 18). The self-assembly of these particlesinto ordered structures creates new possibilities for the fabrication ofnovel materials (19–23). Moreover, advances in synthesis techniqueshave created new capabilities for experimentally investigating howthe shapes of particles can be exploited in their self-assembly (24–26).Here, we experimentally and computationally explore the self-

assembly of colloidal superballs interacting with depletion forces.We find that monolayers of superballs can be tuned to equilibrateinto both their densest known packings—so-called “Λ0” and “Λ1”

lattices (12)—as well as into less dense structures of differentsymmetries depending on an interplay between the subtle featuresof the particle shapes and the size of the depletants. The family ofsuperballs can smoothly interpolate shapes between spheres andcubes (Fig. 1E) and is modeled as

ðxÞm + ðyÞm + ðzÞm ≤ 1, [1]

wherem is the shape parameter. Form= 2, this parameterizationdescribes a purely isotropic sphere. As m is increased, the shapeincreasingly resembles a cube, as shown in Fig. 1. The amorphouscolloidal superballs were prepared via controlled deposition ofsilica on the surface of hematite templates, using a synthetic tech-nique (27) that yields high amounts of monodisperse (3% poly-dispersity) particles. Each batch of particles, which were madefrom the same initial hematite cores, contains superballs of com-parable sizes (∼1.3 μm), but differing shape parameters as a resultof differing amounts of silica precipitated on the surface. Size andshape of superballs were analyzed using scanning electron micros-copy (SEM) and transmission electron microscopy (TEM) micro-graphs (Fig. 1). Analyzing the particle shape from TEM images,we find agreement between the contour of the particle and thesuperball shape as shown in Fig. 1B, in which the red contourscorrespond to the superball fits. More information on the fittingprocedure and shape polydispersity can be found in SI Text.Fig. 1 shows SEM and TEM images of the silica superballs used

for the experiments. Although the particles still possess a distinctcubic symmetry, they have rounded edges whose curvatures areconsistent with superballs of shape parameters m= 2.0, m= 3.0,

Significance

Since antiquity it has been known that particle shape plays anessential role in the symmetry and structure of matter. A fa-miliar example comes from dense packings, such as spheresarranged in a face-centered cubic lattice. For colloidal super-balls, we observe the transition from hexagonal to rhombiccrystals consistent with the densest packings. In addition, wesee the existence of square structures promoted by the pres-ence of depletion attractions in the colloidal system. By using amixture of depletants, one of which is size tunable, we inducesolid-to-solid phase transitions between these phases. Our re-sults introduce a general scenario where particle buildingblocks are designed to assemble not only into their maximumdensity states, but also into depletion-tunable interaction-dependent structures.

Author contributions: L.R., V.S., A.P.P., P.M.C., S.S., and W.T.M.I. designed research; L.R.,V.S., D.J.A., M.D., S.S., andW.T.M.I. performed research; L.R., V.S., D.J.A., P.M.C., M.D., andW.T.M.I. analyzed data; and L.R., V.S., D.J.A., D.J.P., A.P.P., P.M.C., M.D., S.S., and W.T.M.I.wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1Present address: Institute of Physics, University of Amsterdam, Science Park 904, 1098XHAmsterdam, The Netherlands.

2L.R., V.S., and D.J.A. contributed equally to this work.3To whom correspondence may be addressed. Email: L.Rossi@uva.nl, soni@uchicago.edu,or wtmirvine@uchicago.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1415467112/-/DCSupplemental.

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m= 3.5, and m= 3.9. The spherical particles with shape param-eters m= 2 were purchased from Bangs Laboratoires.To perform the experiments, silica superballs were dispersed

in slightly alkaline water (pH = 9) and were stabilized againstaggregation by surface charges. Sodium chloride (10 mM, finalconcentration) was added to the dispersion to screen the chargesand lower the Debye length down to a thickness of about 3 nm,small enough to allow the particles to fully experience their an-isotropic shape. Attractive forces between superballs arise byaddition of depletion agents with gyration radii of Rg = 57 nm,65 nm, 70 nm, 210 nm, 228 nm, and 329 nm. Flat optical capillarieswere filled with aqueous mixtures of superballs and depletants andwere monitored in time with bright-field microscopy. More ex-perimental details as well as information on depletants and samplepreparation can be found in SI Text.At low particle concentration, the superballs first sediment to

the bottom of the capillary where they are attracted to the glasswall by depletion forces. While diffusing in the plane, the par-ticles cluster together into monolayers. Once clusters are formed,time-lapsed images are collected and analyzed. The images showthe appearance of several qualitatively distinct phases (Fig. 2).

The particles are found to arrange into crystallite islands, oftenpossessing grain boundaries, which we separate by orientationand analyze independently. We do not exclude a priori the possi-bility that a cluster does not have a coherent crystal structure.To characterize the structure of each cluster, the positions of

the constituent particles are identified for every time-lapsedimage. The relative positions of nearest neighbors are thencomputed for each particle. For spherical superballs the dis-tribution of these positions are found to be consistent withtriangular lattices (Fig. 2C). For superballs with intermediateshape parameters (2<m<∞), however, the behavior becomesmore interesting. Experimentally, we observe that the particlesoften form canted structures (Fig. 2A) characterized by inter-particle bond angles distinct from 60°, indicative of triangularlattices, and 90°, which are characteristic of square lattices.Recently, the densest packings of superdisks with these in-termediate shape parameters were predicted to fall into twofamilies of lattices, referred to as Λ0 and Λ1 packings (12).Testing the distribution of relative nearest-neighbor positionsin the experiment for consistency with the lattice vectors ofthese structures confirms, for the first time to our knowledge,

A B

C D

E

Fig. 1. (A) SEM images of samplem = 3.9. The particles have a cubic shape with rounded edges. (B–D) TEM micrographs of samples withm = 3.5,m = 3.0, andm = 2.0, respectively. All samples are uniform with a size polydispersity as low as 3%. (Scale bars: 1 μm.) In B the particles are shown with their correspondingsuperball fit highlighted in red (see also Fig. S1). (E , Top) Computer-generated models of colloidal superballs with different shape parameters m. A gradualincrease of the absolute value of the shape parameter from m= 2 (spheres) results in a gradual alteration of the particle shape to resemble more cube-likeparticles. (E, Bottom) TEM images of silica superballs with different m values.

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the observation of an equilibrium Λ1 lattice of superballs in ex-periments (Fig. 2). We also find that, for superballs with thesesame intermediate shape parameters (m= 3.5 and m= 3.9), theequilibrium structure transitions to a square lattice as thedepletant size is decreased, suggesting that the resulting phasesare determined by an interplay between the shape of theparticle and the size ratio q= 2Rg=L between the depletantand the superball.To understand this interplay, we look at the depletion in-

teractions between the superballs. Each superball is surroundedby an exclusion zone of thickness Rg that is unavailable for thecenters of the depletants to occupy. Superball configurations thatminimize the volume excluded from the depletants by over-lapping exclusion zones increase the overall entropy of the sys-tem. To understand the favorability of the three lattices for agiven choice of parameters, we compute the free energy of adepletion-stabilized bound state of a particle for each crystaltype. For a number density n of depletants, this energy is given byU =−nKBTΔVex, where ΔVex is the change in volume excludedwhen a particle is removed from the interior of an otherwisefilled lattice. By computing and comparing ΔVex for the Λ0, Λ1,and square lattices, we estimate which lattice is energetically

favorable for a particular value of m (Fig. 3). In this model, themagnitude of ΔVex, and thus the overall bound state energy, willgenerally scale with Rg. We note that this model neglects theentropy of the superballs. It has been suggested that the role ofrotational entropy of the particles can be significant in stabilizingcanted phases (25), although the relative importance of this ef-fect is debated (13). Fig. 3B shows that, for fixed-sized depletantsand superballs, ΔVex varies smoothly for each lattice type as m isvaried. For a particular combination of m and q, the lattice withthe highest value of ΔVex represents the preferred phase. Usingthis principle, a 2D phase diagram is approximated in Fig. 3C.The interplay between the particle shape and the size ratio qsuggested by this diagram is qualitatively apparent in the ex-perimentally realized structures (Fig. 4).Indeed, the calculations agree with the experimental result

that for sufficiently small depletants and sufficiently large m,square lattices, although they are not the densest packings forany finite value ofm, are preferred. Square lattices occur whenmis large enough such that the overlap in exclusion zones resultingfrom face-to-face contact is considerable and for q small enoughsuch that depletants are able to fit into the interparticle poresmade where the rounded edges of the superballs meet. When theosmotic pressure exerted by a depletant within an interparticlepore is substantial, the cubic phase is stabilized. However, whenintermediate-sized depletants, which can no longer fit into thespaces within the lattice, are dispersed with superballs possessingthese larger values of m (3.5 and 3.9), the densely packed Λ1phase emerges. As the size ratio gets larger, we note the distri-bution of bond angles within a crystallite begins to broaden. Inthe case of m  =   3.9, for the highest size ratio q we tested, thedistribution was too broad to identify the experimental structurewith one of the three lattices, leaving the structure undetermined.

A

B

C

Fig. 2. Representative optical microscope images showing three differentordered structures found in superball samples. (A–C, Right) Histogram of therelative positions of nearest neighbors for each particle in a crystallite (Top)and a histogram of the interparticle bond angles (Bottom) (see also Fig. S2).The structures of the crystallites are characterized by bond angels of 54° (A),90° (B), and 60° (C). Note that the superballs in A and B have the same shape.The different lattice structures in these two samples result from differentdepletant sizes.

A

CD

B

Fig. 3. Two-dimensional predicted diagram for depletion-stabilized super-ball phases. The favorability of each lattice type is determined by calculatingthe bound state energy of a particle. (A) Operationally, the bound stateenergy is found by computing the difference in the excluded volume for aparticular lattice (A, i) and the excluded volume of that lattice when aparticle is removed from the interior (A, ii). (B) Change in excluded volumefor each lattice type with varying m but fixed q = 2Rg/L, where Rg is theradius of gyration of the depletant and L is the diameter of the superball.To illustrate the behavior of ΔVex, the range of m used in this plot is largerthan the experimentally investigated range. Background color indicates thepreferred phase. (C) Two-dimensional phase diagram for experimentalrange of q and m. (D) Difference in ΔVex between two most favorable latticetypes. Near phase boundaries, the phases become degenerate. In addition,for large depletants, the benefit of choosing a particular phase is small (seealso Fig. S3). Recent molecular dynamics simulations (12) of convex super-disks have shown that the critical value of m when the densest packingschange from Λ1 to Λ0 is at m≈ 2.572.

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Whereas our calculations suggest that the Λ1 phase is energeti-cally favorable, Fig. 3D shows that the difference in ΔVex betweenthe lattices with the two highest values becomes negligible forlarge q. This suggests that the energetic benefit of choosing aparticular phase decreases, which is consistent with our observa-tion that the variance in experimental bond angle distributionsincreases for high q.As mentioned previously, spherical superballs form triangular

lattices, which are equivalent to the Λ0 lattice for m= 2. Forsmall deviations from spheres, our calculations suggest that theΛ0 lattice also tends to maximize ΔVex. As the deformation pa-rameter is increased, however, the value of ΔVex for a different

lattice, depending on q, surpasses that of the Λ0 lattice. This canbe seen, for example, in Fig. 3B where the curve representing thesquare lattice intersects the curve representing the Λ0 lattice. Atthis intersection point, the lowest energy state becomes de-generate. Near these regions, the difference in energy betweenthe most favorable lattices is small (Fig. 3D). As a result, we findexperimental structures near phase boundaries fail to conform toa single coherent crystal type. Again, here we find some exper-imental structures are characterized by broad variances in bondangle distributions, which disallow the identification of a par-ticular crystal type. Often, however, although there are in-sufficient statistical data to make a precise classification, we findthe appearance of mixed assortments of crystallites (SI Text) ofboth cubic structures and undetermined, noncubic structureswithin a single sample cell.To more carefully probe the stability of our observed lattices,

we perform idealized simulations of superballs and depletants(SI Text and Fig. S4). We first simulate finite crystallites and findthe results qualitatively agree with experiments (Figs. S5 and S6).A particular choice of initial conditions, however, may influencethe vulnerability of the resulting assembly to fall into kinetictraps. To probe the true stability of our candidate lattices, and toremove surface effects that exist in finite crystallites, we performbulk crystal simulations, using periodic boundary conditions ofeach candidate lattice (Fig. S7). Fig. 4 shows the resulting stablelattices determined from these simulations. The results qualita-tively agree with our excluded volume calculations. It is in-teresting to note that near phase boundaries both Λ1 and squarelattices often can be stable for the same parameters, as suggestedby the appearance of mixed crystallites in experimental struc-tures. In addition, we find that, for m values between 2 and 3,particles often assemble with irregular orientations with respectto their neighbors, consistent with the observation of indeterminateexperimental structures.It is particularly interesting to note that for both experiment

and simulation, we identify different crystalline structures as q is

Fig. 4. Comparison between experimental observations, bulk crystal simu-lations, and calculated phase diagram for superballs at different m and qvalues. Circles indicate the experimental results, open circles indicate simu-lation results, and the background color indicates the predicted phase. Theapproximated phase diagram qualitatively agrees with our experimentaland simulation results.

T=27.5 T=31 T=27

7290

90

T T

deactivated

PEO PNIPAM

activated

PEO PNIPAM

activated

PEOPNIPAM

B

A

Fig. 5. Demonstration of reversible solid–solid phase transition of superballs. (A) Colloidal superballs with shape parameter m = 3.9 dispersed in depletantmixture of PEO and pNIPAM. At 27.5 °C, superballs assemble into a square lattice. At 31 °C, energetic contribution of pNIPAM becomes negligible, whilethat of PEO stays fixed, resulting in the transition into a Λ1 lattice. (B) Simulated phase transition in a bulk crystal of superballs and depletants. A periodiclattice of superballs is simulated along with a mixture of two species of depletants, one with fixed size ratio of q1 = 0.35 (which favors a Λ1 lattice) and asmaller depletant (which favors a square lattice) of size ratio varying from q2 = 0.04 to 0.032. As the smaller depletant is reduced in size, its overall energeticcontribution decreases and the lattice transitions to a Λ1 structure. When the size of the smaller depletant is once again increased, the square lattice onceagain emerges.

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varied for m ≥ 3.5. In principle, it is thus experimentally possibleto use size-variable depletants to reversibly switch the latticestructure within a single sample. To explore this possibility,we use thermosensitive poly(N-isopropylacrylamide) (pNIPAM)microgel spheres as depletants. Although we find inducing astructural transition with pNIPAM depletants alone is difficult(SI Text), we find that, using a bidepletant mixture, we are able toentropically drive a solid–solid transition. This transition dem-onstrates a powerful mechanism in which leveraging differentgeometric features of individual particles enables one to con-trollably and reversibly tune their assembly.To drive this transition, we use superballs with shape param-

eter m= 3.9 and a mixture of polyethylene oxide (PEO) andpNIPAM as depletant. Using the pNIPAM alone, the superballsform square lattices at 25 °C. When they are heated to 29 °C, wefind the overall superball interactions induced by pNIPAM de-crease sufficiently to melt this square lattice (Movie S1). Moreover,when the superballs are dispersed with PEO (molecular weightof 8 M) alone as depletant, we find we are able to stabilize a Λ1lattice of superballs (Movie S2).Using a mixture of the two depletants, however, allows us to

reversibly switch between the two lattice types by varying thetemperature. At room temperature, the interactions induced bythe pNIPAM are activated, and the superballs once again favor asquare lattice. As the temperature is increased, the relative en-ergetic contribution of the pNIPAM depletant decreases, whilethe contribution of the PEO remains the same. Because the PEOdominates the overall energy at high temperatures, the Λ1 latticeemerges. Fig. 5 and Movie S3 demonstrate this reversible solid–solid phase transition.By performing simulations of bidepletant superball dispersions

we provide further evidence of the simple entropic nature of thegeometric mechanism that induces this solid–solid transition.Again we perform periodic simulations of a bulk crystal as well assimulations of finite crystallites. Superballs are dispersed withtwo species of depletant, one with fixed size ratio q1 = 0.35, whichis found to stabilize a Λ1 lattice, and a variable-size depletant withinitial size ratio q2 = 0.04, which is found to stabilize a square lattice.

When dispersed in a mixture with number densities n1 = 24.9L−3

and n2 = 596.8L−3 for the large and small depletant, respectively,superballs with shape parameterm= 4 arrange into a square lattice.Here L is the diameter of the superball. As the smaller depletant isshrunk by 20% at fixed number density, its induced pressure re-mains fixed while its overall energetic contribution is lowered. Wefind, consistent with our experimental observations, that the latticebecomes canted. Upon increasing q2 once again, we find the squarelattice is restored (Figs. S8 and S9 and Movie S4).In this article we have demonstrated the reversible assembly of

the same superball-shaped colloidal particles into both a squarephase and the recently predicted Λ1 phase. We show depletantsize can be used to tune interparticle interactions. As a result,both particle shape and depletant size are used to determine theresulting phases. By mixing large depletants and small thermo-sensitive depletants we demonstrate a fully reversible solid-to-solidtransition between square and Λ1 superball phases. The sensitivityof the assembled phase to a fine feature of the particle shape,combined with a mechanism to reversibly activate a depletant onthat scale, demonstrates that depletants can be used to tune in-teractions. These results create previously unidentified opportuni-ties for controlling the reversible self-assembly of colloidalparticles and controlling phases, for example through solid-to-solid phase transitions.

ACKNOWLEDGMENTS. Prof. Nigel B. Wilding, Dr. Krassimir P. Velikov,Dr. Andrei Petukhov, Prof. Willem Kegel, Dr. Ra Ni, Dr. Frank Smallenburg,Theodore Hueckel, Prof. S. R. Nagel, and Prof. T. Witten are thanked formany useful discussions. Prof. Dirk Aarts is thanked for kindly providing theXanthan polymer. We acknowledge the Materials Research and EngineeringCenters (MRSEC) Shared Facilities at The University of Chicago for the use oftheir instruments, and NSF MRI 1229456. This work was supported by theNational Science Foundation MRSEC Program at The University of Chicago(NSF DMR-MRSEC 1420709). W.T.M.I. further acknowledges support fromthe A. P. Sloan Foundation through a Sloan fellowship, and the PackardFoundation through a Packard fellowship. L.R. and A.P.P. acknowledgeAgentschap NL for financial support through Grant FND07002. Engineeringand Physical Sciences Research Council (EPSRC) is acknowledged for supportto D.J.A. through Grant EP/I036192/1. P.M.C. acknowledges support fromNASA (NNX08AK04G).

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